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Positive results of dark matter searches in experiments DAMA/NaI andDAMA/LIBRA taken together with negative results of other groups can implynontrivial particle physics solutions for cosmological dark matter. Stableparticles with charge -2 bind with primordial helium in O-helium "atoms" (OHe),representing a specific Warmer than Cold nuclear-interacting form of darkmatter. Slowed down in the terrestrial matter, OHe is elusive for directmethods of underground Dark matter detection like those used in CDMSexperiment, but its low energy binding with nuclei can lead to annualvariations of energy release in the interval of energy 2-6 keV in DAMA/NaI andDAMA/LIBRA experiments. Schrodinger equation for system of nucleus and OHe issolved for a spherically symmetrical potential, formed by the Yukawa tail ofnuclear scalar isoscalar attraction potential, acting on He beyond the nucleus,and dipole Coulomb repulsion between the nucleus and OHe at distances from thenuclear surface, smaller than the size of OHe. The values of coupling strengthand mass of meson, mediating scalar isoscalar, are rather uncertain. Within theuncertainties of parameters of nuclear potential we find a range of theseparameters, at which the sodium and/or iodine nuclei have a few keV bindingenergy with OHe. At nuclear parameters, reproducing DAMA results, the energyrelease predicted for detectors with chemical content other than NaI differ inthe most cases from the one in DAMA detector. In particular, it is shown thatin the case of CDMS germanium state has binding energy with OHe beyond therange of 2-6 keV and its formation should not lead to ionization in the energyrange of DAMA signal. (abridged)

Low energy binding of composite dark matter with nuclei as a solution for the puzzles of dark matter searches